## Model

In our design, integrases compute the output of the logic gates. Integrases allow flipping one fragment of DNA. Therefore, they are of central importance to our design. However, their characterization in literature^{[9]} is incomplete. In particular, quantitative insight into dimerization rates and DNA-binding rates is lacking. Such data is however necessary in order to be able to set up a mathematical model to describe the overall Mosai*coli* process. Thus, we decided to estimate the missing parameters from published from published experiments based on a model that we developed ourselves.

### Chemical Species

Name
| Description |
---|---|

Bxb1
| Serine integrase that can fold into two conformations - Bxb1a and Bxb1b. We chose to use a common connotation for both conformations - Bxb1. |

ΦC31
| Serine integrase that can fold into two conformations - ΦC31a and ΦC31b. We chose to use a common connotation for both conformations - ΦC31. |

DBxb1
| Dimerized form of Bxb1. We chose to use a common connotation for both homodimers, DBxb1a and DBxb1b. |

DΦC31
| Dimerized form of ΦC31. We chose to use a common connotation for both homodimers, DΦC31a and DΦC31b. |

### Modeling DNA-binding sites

Each dimer of integrases can specifically bind to a DNA binding site. As the flipping is irreversible, these DNA binding sites can be in three possible states:

- SI
_{IntegraseName}: inactive DNA binding site. No dimer is bound to this site, which has never been flipped.

- SA
_{IntegraseName}: active DNA binding site. A dimer is bound to this site.

- SF
_{IntegraseName}: flipped DNA binding site. This site has been irreversibly flipped.

### Reactions

- For Bxb1

$$ \begin{align} Bxb1 + Bxb1 &\leftrightarrow DBxb1 \\ DBxb1 + SI_{Bxb1} & \leftrightarrow SA_{Bxb1}\\ Bxb1 &\rightarrow \\ DBxb1 &\rightarrow \end{align}$$

- For ΦC31

\begin{align} \phi C31 + \phi C31 &\leftrightarrow D\phi C 31 \\ D\phi C 31 + SI_{\phi C31} & \leftrightarrow SA_{\phi C31}\\ \phi C31 &\rightarrow \\ D\phi C31 &\rightarrow \end{align}

### Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations for Bxb1.

$$\frac{d[Bxb1]}{dt}=-2 k_{DBxb1}[Bxb1]^2+ 2 k_{-DBxb1}[DBxb1]-d_{Bxb1}[Bxb1]$$

$$\frac{d[DBxb1]}{dt}=-k_{SABxb1}[DBxb1][SI_{Bxb1}]+k_{-SABxb1}[SA_{Bxb1}]+k_{DBxb1}[Bxb1]^2-k_{-DBxb1}[DBxb1]-d_{DBxb1}[DBxb1]$$

$$\frac{d[SA_{Bxb1}]}{dt}=k_{SABxb1}[DBxb1][SI_{Bxb1}]-k_{-SABxb1}[SA_{Bxb1}]$$

Replacing every occurence of Bxb1 by ΦC31 gives the set of differential equations for ΦC31.